(1) Field of Invention
The present invention relates to a system for accurately reconstructing analog signals and, more particularly, to a system for accurately reconstructing analog signals in a continuous sparsifying domain.
(2) Description of Related Art
Compressive sensing is a signal processing technique for efficiently acquiring and reconstructing a signal by finding solutions to underdetermined linear systems. Compressive sensing is able to reconstruct signals with measurements far below the Nyquist rate, under the assumption that the signal is sparse (e.g., the number of frequencies is small). Since the compressive sensing theory is developed from the discrete signal domain, it is limited when dealing with real analog signals with continuous frequency band.
The compressive sensing (CS) theory was set forth by Donoho in “Compressed sensing” in IEEE Trans. on Information Theory, 52:1289-1306, 2006 and Candes in “Near optimal signal recovery from random projection: universal encoding strategies?” in IEEE Trans. on Information Theory, 52:5406-5425, 2006, which are both hereby incorporated by reference as though fully set forth herein. The CS theory states that it is possible to reconstruct signals with measurements far below the Nyquist rate if the signals are sparse in some known transform domain. There has been a substantial amount of research on refining the theoretical results (especially towards the conditions on measurements matrices), improving the efficiency of reconstruction algorithms, and turning the CS theory into practical applications in a wide range of areas. One application is to measure and reconstruct frequency-sparse signals that occur in communications (i.e., the number of frequencies is small). However, since the formulation of compressive sensing is discrete, it is only able to properly deal with a discrete sparsifying domain (i.e., frequencies on the discretized grid), which is not realistic in practice.
Further, Duane and Baraniuk proposed in “Spectral Compressive Sensing” in Applied and Computational Harmonic Analysis, 2012, a suite of spectral CS (SCS) recovery algorithms for arbitrary frequency-sparse signals. This reference, hereinafter referred to as the Duarte and Baraniuk reference, is hereby incorporated by reference as though fully set forth herein. The method described in the Duarte and Baraniuk reference uses an over-sampled discrete Fourier Transform (DFT) frame together with a coherence-inhibiting structured signal model. The method was demonstrated to outperform current state-of-the-art CS algorithms based on the DFT and classical sinusoid parameter estimation algorithms. However, in terms of accuracy of parameter estimation, there is room for improvement.
Thus, a continuing need exists for a method that is able to deal with a continuous sparsifying domain and accurately reconstruct signals with arbitrary frequencies from non-uniform samples with sampling rates much lower than the Nyquist rate.